Comparing upper broadcast domination and boundary independence broadcast numbers of graphs

Mynhardt, Kieka; Neilson, Linda

doi:10.22108/toc.2023.127904.1836

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	Comparing upper broadcast domination and boundary independence broadcast numbers of graphs
Transactions on Combinatorics
دوره 13، شماره 1، خرداد 2024، صفحه 105-126 اصل مقاله (630.59 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/toc.2023.127904.1836
نویسندگان
Kieka Mynhardt^* ¹؛ Linda Neilson²
¹Department of Mathematics and Statistics, University of Victoria, P. O.Box 3800, Victoria, Canada
²Department of Adult Basic Education, Vancouver Island University Nanaimo,Canada
چکیده
A broadcast on a nontrivial connected graph $G=(V,E)$ is a function $f:V\rightarrow\{0, 1,\dots,d\}$, where $d=\operatorname{diam}(G)$, such that $f(v)\leq e(v)$ (the eccentricity of $v$) for all $v\in V$. The weight of $f$ is $\sigma(f)={\textstyle\sum_{v\in V}} f(v)$. A vertex $u$ hears $f$ from $v$ if $f(v)>0$ and $d(u,v)\leq f(v)$. A broadcast $f$ is dominating if every vertex of $G$ hears $f$. The upper broadcast domination number of $G$ is $\Gamma_{b}(G)=\max\left\{ \sigma(f):f\text{ is a minimal dominating broadcast of }G\right\}.$ A broadcast $f$ is boundary independent if, for any vertex $w$ that hears $f$ from vertices $v_{1},\ldots,v_{k},\ k\geq2$, the distance $d(w,v_{i})=f(v_{i})$ for each $i$. The maximum weight of a boundary independent broadcast is the boundary independence broadcast number $\alpha_{\operatorname{bn}}(G)$. We compare $\alpha_{\operatorname{bn}}$ to $\Gamma_{b}$, showing that neither is an upper bound for the other. We show that the differences $\Gamma _{b}-\alpha_{\operatorname{bn}}$ and $\alpha_{\operatorname{bn}}-\Gamma_{b}$ are unbounded, the ratio $\alpha_{\operatorname{bn}}/\Gamma_{b}$ is bounded for all graphs, and $\Gamma_{b}/\alpha_{\operatorname{bn}}$ is bounded for bipartite graphs but unbounded in general.
کلیدواژه‌ها
broadcast domination؛ broadcast independence؛ hearing independent broadcast؛ boundary independent broadcast

مراجع
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Comparing upper broadcast domination and boundary independence broadcast numbers of graphs